On some homogeneization problems from shallow water theory

Didier Bresch (1, 2) , David Gerard-Varet (3)

1 LAMA - Laboratoire de Mathématiques
2 LMC - IMAG - Laboratoire de Modélisation et Calcul
3 DMA - Département de Mathématiques et Applications - ENS Paris

This note is devoted to the effect of topography on geophysical flows. We consider two models derived from shallow water theory: the quasigeostrophic equation and the lake equation. Small scale variations of topography appear in these models through a periodic function, of small wavelength $\varepsilon$. The asymptotic limit as $\varepsilon$ goes to zero reveals homogenization problems in which the cell and averaged equations are both nonlinear. In the spirit of article [P.-L. Lions, N. Masmoudi, Homogenization of the Euler system in a 2D porous medium, J. Math. Pures Appl. (9) 84 (1) (2005) 1–20], we derive rigorously the limit systems, through the notion of two-scale convergence.

Mots clés: Shallow water equations; Two-scale convergence; Homogenization

Domaines: Equations aux dérivées partielles [math.AP]

Didier Bresch, David Gerard-Varet. On some homogeneization problems from shallow water theory. Applied Mathematics Letters, 2006, 20 (5), pp.505-510. ⟨10.1016/j.aml.2006.05.018⟩. ⟨hal-00385846⟩